A020714 -id:A020714 - cp's OEIS frontend

A375204 Record values in A375202.

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 512, 576, 640, 768, 1024, 1152, 1280, 1536, 2048, 2304, 2560, 3072, 4096, 4608, 5120, 6144, 8192, 9216, 10240, 12288, 16384

Offset: 1

Robert Israel, Oct 15 2024

Numbers k such that k = A375202(m) for some m such that A375202(j) < k for all j < m.

Conjectures: All powers of 2 (A000079), 3*(powers of 2) (A007283) and 5*(powers of 2) (A020714) are terms. All terms are 5-smooth (A051037). - Chai Wah Wu, Oct 16 2024

			a(3) = 2 because A375202(12) = 2 and A375202(j) <= 1 for j < 12.

Mathematics StackExchange, Large smallest square in a sum of three squares?

Cf. A000079, A007283, A020714, A051037, A375202, A375203.

f:= proc(n) local q, x, y, z;
  if n/4^padic:-ordp(n, 4) mod 8 = 7 then return -1 fi;
  for x from 0 while 3*x^2 <= n do
    if [isolve(y^2 + z^2 = n - x^2)] <> [] then return x fi
  od;
end proc:
V:= NULL:count:= 0: m:= -1;
for i from 0 while count < 39 do
  v:= f(i);
  if v > m then
    V:= V, v; m:= v; count:=count+1
  fi
od:
V;

from itertools import count, islice
from math import isqrt
from sympy import factorint
def A375204_gen(): # generator of terms
    c = -1
    for n in count(0):
        v = (~n & n-1).bit_length()
        if v&1 or n>>v&7!=7:
            a = next(x for x in range(isqrt(n//3)+1) if not any(e&1 and p&3==3 for p, e in factorint(n-x**2).items()))
            if a>c:
                yield a
                c = a
A375204_list = list(islice(A375204_gen(),20)) # Chai Wah Wu, Oct 16 2024

a(n) = A375292(A375203(n)).

a(35)-a(48) from Chai Wah Wu, Oct 16 2024

a(49)-a(52) from Chai Wah Wu, Oct 17 2024

A168330 Period 2: repeat [3, -2].

Original entry on oeis.org

3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2

Offset: 1

Klaus Brockhaus, Nov 23 2009

Interleaving of A010701 and -A007395.
Binomial transform of 3 followed by a signed version of A020714.
Inverse binomial transform of 3 followed by A000079.
A084964 without first two terms gives partial sums.

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A168309 (repeat 4, -3), A010701 (all 3's sequence), A007395 (all 2's sequence), A010716 (all 5's sequence), A020714 (5*2^n), A000079 (powers of 2), A084964 (follow n+2 by n).

Magma
```
&cat[[3,-2]: n in [1..42]];
```

[n eq 1 select 3 else -Self(n-1)+1:n in [1..84]];

[(-5*(-1)^n+1)/2: n in [1..100]]; // Vincenzo Librandi, Jul 19 2016

LinearRecurrence[{0, 1}, {3, -2}, 25] (* G. C. Greubel, Jul 18 2016 *)
PadRight[{},120,{3,-2}] (* Harvey P. Dale, Oct 05 2016 *)

a(n)=3-n%2*5 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = (-5*(-1)^n + 1)/2.
a(n+1) - a(n) = 5*(-1)^n.
a(n) = -a(n-1) + 1 for n > 1; a(1) = 3.
a(n) = a(n-2) for n > 2; a(1) = 3, a(2) = -2.
G.f.: x*(3 - 2*x)/((1-x)*(1+x)).
a(n) = A049071(n). - R. J. Mathar, Nov 25 2009
E.g.f.: (1/2)*(1 - exp(-x))*(5 + exp(x)). - G. C. Greubel, Jul 18 2016

A199112 a(n) = 10*3^n+1.

Original entry on oeis.org

11, 31, 91, 271, 811, 2431, 7291, 21871, 65611, 196831, 590491, 1771471, 5314411, 15943231, 47829691, 143489071, 430467211, 1291401631, 3874204891, 11622614671, 34867844011, 104603532031, 313810596091, 941431788271, 2824295364811

Offset: 0

Vincenzo Librandi, Nov 04 2011

The inverse binomial transform is 11, 20, 40, 80, 160,... closely related to A020714. - R. J. Mathar, Apr 07 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A020714.

Magma
```
[10*3^n+1: n in [0..30]];
```

10*3^Range[0,30]+1 (* or *) LinearRecurrence[{4,-3},{11,31},30] (* Harvey P. Dale, Dec 07 2011 *)

a(n) = 3*a(n-1)-2.
a(n) = 4*a(n-1)-3*a(n-2).
G.f.: (11-13*x)/((1-x)*(1-3*x)). - Bruno Berselli, Nov 04 2011

A318163 a(0) = a(3) = 0, a(1) = a(2) = 1; for n >= 2, a(2n) = -a(n-1) and a(2n+1) = -a(n-1)-a(n).

Original entry on oeis.org

0, 1, 1, 0, -1, -2, -1, -1, 0, 1, 1, 3, 2, 3, 1, 2, 1, 1, 0, -1, -1, -2, -1, -4, -3, -5, -2, -5, -3, -4, -1, -3, -2, -3, -1, -2, -1, -1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, -1, -1, -2, -1, -3, -2, -3, -1, -4, -3, -5, -2, -5, -3, -4, -1, -6, -5, -9, -4, -11, -7

Offset: 0

Altug Alkan, Aug 19 2018

Inspired by A002487.
Alternatively, a(0) = 0, a(1) = 1; for n >= 1, a(2*n) = a(2*n-1) - a(2*n-2), a(2*n+1) = a(2*n) - a(n). Note that if b(0) = 0, b(1) = 1; for n >= 1, b(2*n) = b(2*n-1) - b(n), b(2*n+1) = b(2*n) - b(2*n-1), then b(n) + A213369(n+1) = 0 for all n >= 1.
The main block structure of this sequence is described by A020714.

Altug Alkan, Table of n, a(n) for n = 0..20480
Altug Alkan, A scatterplot of a(n) for n <= 5*2^12
Altug Alkan, A scatterplot of first differences of a(n) for n <= 5*2^13

Cf. A002487, A020944, A213369, A303404.

a[0]=a[3]=0; a[1]=a[2]=1; a[n_] := a[n] = If[EvenQ[n], -a[n/2-1], -a[(n-1)/2 - 1] - a[(n-1)/2]]; Array[a, 101, 0] (* Giovanni Resta, Aug 27 2018 *)

a = vector(100); print1(0", "); for(k=1, #a, print1 (a[k]=if(k<=2,1, my (n=k\2); if (k%2==0, -a[n-1], a[2*n]-a[n]))", "));

a(5*2^k-2) = 0 for all k >= 0.

A348363 The 1's in the binary expansion of a(n) encode the distances between the 1's in the binary expansion of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 15, 3, 15, 7, 15, 1, 17, 9, 27, 5, 21, 15, 31, 3, 27, 15, 31, 7, 31, 15, 31, 1, 33, 17, 51, 9, 45, 27, 63, 5, 45, 21, 63, 15, 47, 31, 63, 3, 51, 27, 59, 15, 63, 31, 63, 7, 63, 31, 63, 15, 63, 31, 63, 1, 65, 33, 99, 17, 85, 51

Offset: 0

Rémy Sigrist, Oct 15 2021

The bit 2^d is set in a(n) iff for some e >= 0, the bits 2^e and 2^(e+d) are set in n.
This sequence has similarities with A067398; here we take the absolute differences, there the sums, of indices of 1's in binary expansions.
All terms are odd, except a(0) = 0.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     1      10          1
   3     3      11         11
   4     1     100          1
   5     5     101        101
   6     3     110         11
   7     7     111        111
   8     1    1000          1
   9     9    1001       1001
  10     5    1010        101
  11    15    1011       1111
  12     3    1100         11
  13    15    1101       1111
  14     7    1110        111
  15    15    1111       1111

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Colored scatterplot of the first 2^20 terms (where the color is function of the 2-adic valuation of n, upper red pixels correspond to odd n's)
Index entries for sequences related to binary expansion of n

Cf. A000079, A007283, A020714, A064896, A067398.

{0}~Join~Array[Total[2^Append[Union@ Abs[Subtract @@@ Permutations[1 + Length[#] - Position[#, 1][[All, 1]] &@ IntegerDigits[#, 2], {2}]], 0]] &, 70] (* Michael De Vlieger, Oct 16 2021 *)

a(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n-=2^b[k]=valuation(n, 2);); my (p=setbinop((i,j)->abs(i-j), b)); sum (k=1, #p, 2^p[k]) }

def a(n):
    locs = [e for e in range(n.bit_length()) if 1 & (n>>e)]
    diffs = set(abs(e1-e2) for i, e1 in enumerate(locs) for e2 in locs[i:])
    return sum(2**d for d in diffs)
print([a(n) for n in range(71)]) # Michael S. Branicky, Oct 16 2021

a(2*n) = a(n).
a(n) = n iff n = 0 or n belongs to A064896.
a(n) = 1 iff n is a power of 2 (A000079).
a(n) = 3 iff n belongs to A007283.
a(n) = 5 iff n belongs to A020714.
a(n) AND n = n for any odd number n (where AND denotes the bitwise AND operator).

A376065 Orders k of groups G such that Inn(G) is isomorphic to Z(Aut(G)) for at least one G of order k.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 32, 48, 64, 72, 96, 128, 144, 192, 216, 243, 256, 288, 320, 384, 432, 486, 512, 576, 640, 648, 729, 768, 864

Offset: 1

Miles Englezou, Sep 07 2024

Inn(G) is the inner automorphism group of G and Z(Aut(G)) is the center of the automorphism group of G.
A group G for which Inn(G) = Z(Aut(G)) allows for a natural construction of Aut(Aut(G)) via the homomorphism f: Aut(G) -> Aut(Aut(G)) which maps Aut(G) to Inn(Aut(G)) = Aut(G)/Z(Aut(G)) in the same way that G is mapped to Inn(G) = G/Z(G). Furthermore Inn(Aut(G)) = Out(G) (the outer automorphism group), and we have an exact sequence of homomorphisms 1 -> G -> Aut(G) -> Aut(Aut(G)) -> 1. Each term a(n) is thus the order of a group which allows for this particular construction of Aut(Aut(G)).
The diagram of homomorphisms is as follows:
             Aut(Aut(G)) --> Out(Aut(G))
            /      ^          /
           /       |         /
     Aut(G)  -->  Inn(Aut(G)) (= Aut(G)/Z(Aut(G)) = Out(G))
    /  ^         /
   /   |        /
 G --> Z(Aut(G)) (= Inn(G))
 ^    /
 |   /
 Z(G)
A000079, A007283(m) for m >= 3, and A020714(r) for r >= 6, are subsequences. See the Miles Englezou link for proofs. In the link it is also shown that the method of proof used to determine that A007283(m) and A020714(r) are subsequences is limited to Fermat primes (A019434) and therefore cannot be used to determine whether 2^s*p is a subsequence for every prime p.

			24 is a term since for G = C3 x D8, Inn(G) = Z(Aut(G)) = C2 x C2, and G has order 24.

Miles Englezou, Proofs of subsequences

Cf. A000079, A007283, A020714, A019434.

S:=[];
for n in [1..500] do
    for i in [1..NrSmallGroups(n)] do
        G:=SmallGroup(n,i);
        Aut:=AutomorphismGroup(G);
        Inn:=InnerAutomorphismsAutomorphismGroup(Aut);
        if IsomorphismGroups(Centre(Aut),Inn)<>fail then
            S:=Concatenation(S,[n]);
            break;
        fi;
    od;
od;
Print(S);

A381793 Smallest k>1 such that 10k^(52^n)+1 is prime.

Original entry on oeis.org

6, 11, 649, 792, 1034, 12386, 21813, 87318, 35387, 207339, 67958

Offset: 0

Jakub Buczak, Mar 07 2025

			a(0) = 6, because 10*6^(5*2^0)+1 equals 77761 which is prime.

Cf. A020714, A089319.

Cf. A381815.

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(2) if isprime(k**(5*(2**n)) * 10 + 1))

a(7)-a(10) from Michael S. Branicky, Mar 07 2025

A159020 a(0)=11; a(n) = a(n-1) + floor(sqrt(a(n-1))), n > 0.

Original entry on oeis.org

11, 14, 17, 21, 25, 30, 35, 40, 46, 52, 59, 66, 74, 82, 91, 100, 110, 120, 130, 141, 152, 164, 176, 189, 202, 216, 230, 245, 260, 276, 292, 309, 326, 344, 362, 381, 400, 420, 440, 460, 481, 502, 524, 546, 569

Offset: 0

Philippe Deléham, Apr 02 2009

nonn

Row 2 in square array A159016.
This sequence contains an infinite number of squares. - Philippe Deléham, Apr 04 2009
The squares in this sequence are A020714(k)^2, k=0,1,2,.... - Vincenzo Librandi, Apr 20 2009 [clarified by R. J. Mathar, Dec 05 2010]

Cf. A028392, A020714.

A228305 a(1) = 3; for n >= 1, a(2n) = 2^(n+1), a(2n+1) = 5*2^(n-1).

Original entry on oeis.org

3, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 2048, 2560, 4096, 5120, 8192, 10240, 16384, 20480, 32768, 40960, 65536, 81920, 131072, 163840, 262144, 327680, 524288, 655360, 1048576, 1310720, 2097152, 2621440, 4194304, 5242880

Offset: 1

Arkadiusz Wesolowski, Aug 20 2013

Union of A020714 and A198633.
Essentially the same as A094958.
For every n, a(1)^3 + a(2)^3 + a(3)^3 + ... + a(2*n-1)^3 is a cube.

			a(9) = 40 because it is equal to 5*2^(4-1).

Wikipedia, Cube (algebra)
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A020714, A094958, A121451, A164682, A198633.

[n le 3 select n+2 else 2*Self(n-2) : n in [1..43]];

CoefficientList[Series[(3 + 4 x - x^2)/(1 - 2 x^2), {x, 0, 50}], x] (* Bruno Berselli, Aug 20 2013 *)

r=43; print1(3); print1(", "); for(n=2, r, if(bitand(n, 1), print1(5*2^((n-3)/2)), print1(2^(n/2+1))); print1(", "));

a(n) = ceiling((9 - (- 1)^n)*2^(floor(n/2) - 2)).
a(n) = n + 2 for n <= 3; a(n) = 2*a(n-2) for n > 3.
From Bruno Berselli, Aug 20 2013: (Start)
G.f.: x*(3+4*x-x^2)/(1-2*x^2).
a(n) = (16-(8-5*r)*(1-(-1)^n))*r^(n-6) for n>1, r=sqrt(2). (End)
E.g.f.: (8*cosh(sqrt(2)*x) + 5*sqrt(2)*sinh(sqrt(2)*x) + 2*x - 8)/4. - Stefano Spezia, Apr 09 2025

A268896 Start at a(0)=1. a(n) = a(n-1)+2 if n == 1,2 (mod 3) and a(n)=a(n-1)+a(n-3) if n == 0 (mod 3).

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 16, 18, 20, 36, 38, 40, 76, 78, 80, 156, 158, 160, 316, 318, 320, 636, 638, 640, 1276, 1278, 1280, 2556, 2558, 2560, 5116, 5118, 5120, 10236, 10238, 10240, 20476, 20478, 20480, 40956, 40958

Offset: 0

Ravesh Sukhram, Feb 27 2016

See Mathematica section for an explicit formula for the n-th term. - Benedict W. J. Irwin, May 30 2016

Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-2).

Simplify[Table[1/6 (10 (2^n)^(1/3) + 4 (-3 + 5 2^(n/3)) Cos[(2 n Pi)/3] + 5 2^((4 + n)/3)Sin[(n Pi)/3] (Sqrt[3] (-1 + 2^(1/3)) Cos[(n Pi)/3] + (1 + 2^(1/3)) Sin[(n Pi)/3]) - 4 (3 + Sqrt[3] Sin[(2 n Pi)/3])), {n, 0, 20}]] (* Benedict W. J. Irwin, May 30 2016 *)

G.f.: ( 1+3*x+5*x^2+3*x^3-x^4-5*x^5 ) / ( (x-1)*(2*x^3-1)*(1+x+x^2) ). - R. J. Mathar, Apr 16 2016

a(3n) = A048487(n). a(3n+1) = A131051(n+1). a(3n+2)=A020714(n). - R. J. Mathar, Apr 16 2016

cp's OEIS Frontend

A375204 Record values in A375202.

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Maple

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A168330 Period 2: repeat [3, -2].

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Magma

Magma

Magma

Mathematica

PARI

Formula

A199112 a(n) = 10*3^n+1.

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Magma

Mathematica

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A318163 a(0) = a(3) = 0, a(1) = a(2) = 1; for n >= 2, a(2*n) = -a(n-1) and a(2*n+1) = -a(n-1)-a(n).

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Mathematica

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A348363 The 1's in the binary expansion of a(n) encode the distances between the 1's in the binary expansion of n.

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Crossrefs

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Mathematica

PARI

Python

Formula

A376065 Orders k of groups G such that Inn(G) is isomorphic to Z(Aut(G)) for at least one G of order k.

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GAP

A381793 Smallest k>1 such that 10*k^(5*2^n)+1 is prime.

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A318163 a(0) = a(3) = 0, a(1) = a(2) = 1; for n >= 2, a(2n) = -a(n-1) and a(2n+1) = -a(n-1)-a(n).

A381793 Smallest k>1 such that 10k^(52^n)+1 is prime.

A228305 a(1) = 3; for n >= 1, a(2n) = 2^(n+1), a(2n+1) = 5*2^(n-1).